3.35 \(\int \frac{(c+d x^2)^5}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=196 \[ \frac{d^3 x \left (6 a^2 d^2-15 a b c d+10 b^2 c^2\right )}{b^5}+\frac{\left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{11/2}}+\frac{x (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left (a+b x^2\right )}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{x (b c-a d)^5}{4 a b^5 \left (a+b x^2\right )^2}+\frac{d^5 x^5}{5 b^3} \]

[Out]

(d^3*(10*b^2*c^2 - 15*a*b*c*d + 6*a^2*d^2)*x)/b^5 + (d^4*(5*b*c - 3*a*d)*x^3)/(3*b^4) + (d^5*x^5)/(5*b^3) + ((
b*c - a*d)^5*x)/(4*a*b^5*(a + b*x^2)^2) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*x)/(8*a^2*b^5*(a + b*x^2)) + ((b*c -
a*d)^3*(3*b^2*c^2 + 14*a*b*c*d + 63*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(11/2))

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Rubi [A]  time = 0.226919, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {390, 1157, 385, 205} \[ \frac{d^3 x \left (6 a^2 d^2-15 a b c d+10 b^2 c^2\right )}{b^5}+\frac{\left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{11/2}}+\frac{x (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left (a+b x^2\right )}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{x (b c-a d)^5}{4 a b^5 \left (a+b x^2\right )^2}+\frac{d^5 x^5}{5 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^5/(a + b*x^2)^3,x]

[Out]

(d^3*(10*b^2*c^2 - 15*a*b*c*d + 6*a^2*d^2)*x)/b^5 + (d^4*(5*b*c - 3*a*d)*x^3)/(3*b^4) + (d^5*x^5)/(5*b^3) + ((
b*c - a*d)^5*x)/(4*a*b^5*(a + b*x^2)^2) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*x)/(8*a^2*b^5*(a + b*x^2)) + ((b*c -
a*d)^3*(3*b^2*c^2 + 14*a*b*c*d + 63*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(11/2))

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (c+d x^2\right )^5}{\left (a+b x^2\right )^3} \, dx &=\int \left (\frac{d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right )}{b^5}+\frac{d^4 (5 b c-3 a d) x^2}{b^4}+\frac{d^5 x^4}{b^3}+\frac{(b c-a d)^3 \left (b^2 c^2+3 a b c d+6 a^2 d^2\right )+5 b d (b c-a d)^3 (b c+3 a d) x^2+10 b^2 d^2 (b c-a d)^3 x^4}{b^5 \left (a+b x^2\right )^3}\right ) \, dx\\ &=\frac{d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac{d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac{d^5 x^5}{5 b^3}+\frac{\int \frac{(b c-a d)^3 \left (b^2 c^2+3 a b c d+6 a^2 d^2\right )+5 b d (b c-a d)^3 (b c+3 a d) x^2+10 b^2 d^2 (b c-a d)^3 x^4}{\left (a+b x^2\right )^3} \, dx}{b^5}\\ &=\frac{d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac{d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac{d^5 x^5}{5 b^3}+\frac{(b c-a d)^5 x}{4 a b^5 \left (a+b x^2\right )^2}-\frac{\int \frac{-(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+23 a^2 d^2\right )-40 a b d^2 (b c-a d)^3 x^2}{\left (a+b x^2\right )^2} \, dx}{4 a b^5}\\ &=\frac{d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac{d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac{d^5 x^5}{5 b^3}+\frac{(b c-a d)^5 x}{4 a b^5 \left (a+b x^2\right )^2}+\frac{(b c-a d)^4 (3 b c+17 a d) x}{8 a^2 b^5 \left (a+b x^2\right )}+\frac{\left ((b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right )\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^2 b^5}\\ &=\frac{d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac{d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac{d^5 x^5}{5 b^3}+\frac{(b c-a d)^5 x}{4 a b^5 \left (a+b x^2\right )^2}+\frac{(b c-a d)^4 (3 b c+17 a d) x}{8 a^2 b^5 \left (a+b x^2\right )}+\frac{(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.122091, size = 196, normalized size = 1. \[ \frac{d^3 x \left (6 a^2 d^2-15 a b c d+10 b^2 c^2\right )}{b^5}+\frac{\left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{11/2}}+\frac{x (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left (a+b x^2\right )}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{x (b c-a d)^5}{4 a b^5 \left (a+b x^2\right )^2}+\frac{d^5 x^5}{5 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^5/(a + b*x^2)^3,x]

[Out]

(d^3*(10*b^2*c^2 - 15*a*b*c*d + 6*a^2*d^2)*x)/b^5 + (d^4*(5*b*c - 3*a*d)*x^3)/(3*b^4) + (d^5*x^5)/(5*b^3) + ((
b*c - a*d)^5*x)/(4*a*b^5*(a + b*x^2)^2) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*x)/(8*a^2*b^5*(a + b*x^2)) + ((b*c -
a*d)^3*(3*b^2*c^2 + 14*a*b*c*d + 63*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(11/2))

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Maple [B]  time = 0.013, size = 484, normalized size = 2.5 \begin{align*}{\frac{{d}^{5}{x}^{5}}{5\,{b}^{3}}}-{\frac{{d}^{5}{x}^{3}a}{{b}^{4}}}+{\frac{5\,{d}^{4}{x}^{3}c}{3\,{b}^{3}}}+6\,{\frac{{a}^{2}{d}^{5}x}{{b}^{5}}}-15\,{\frac{ac{d}^{4}x}{{b}^{4}}}+10\,{\frac{{c}^{2}{d}^{3}x}{{b}^{3}}}+{\frac{17\,{a}^{3}{x}^{3}{d}^{5}}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{65\,{a}^{2}{x}^{3}c{d}^{4}}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{45\,a{x}^{3}{c}^{2}{d}^{3}}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{25\,{x}^{3}{c}^{3}{d}^{2}}{4\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,{x}^{3}{c}^{4}d}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}+{\frac{3\,b{x}^{3}{c}^{5}}{8\, \left ( b{x}^{2}+a \right ) ^{2}{a}^{2}}}+{\frac{15\,x{a}^{4}{d}^{5}}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{55\,{a}^{3}cx{d}^{4}}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,{a}^{2}{c}^{2}x{d}^{3}}{4\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,a{c}^{3}x{d}^{2}}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,x{c}^{4}d}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,x{c}^{5}}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{63\,{a}^{3}{d}^{5}}{8\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{175\,{a}^{2}c{d}^{4}}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{75\,a{c}^{2}{d}^{3}}{4\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{15\,{c}^{3}{d}^{2}}{4\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{c}^{4}d}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,{c}^{5}}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^5/(b*x^2+a)^3,x)

[Out]

1/5*d^5*x^5/b^3-d^5/b^4*x^3*a+5/3*d^4/b^3*x^3*c+6*d^5/b^5*a^2*x-15*d^4/b^4*a*c*x+10*d^3/b^3*c^2*x+17/8/b^4/(b*
x^2+a)^2*a^3*x^3*d^5-65/8/b^3/(b*x^2+a)^2*a^2*x^3*c*d^4+45/4/b^2/(b*x^2+a)^2*a*x^3*c^2*d^3-25/4/b/(b*x^2+a)^2*
x^3*c^3*d^2+5/8/(b*x^2+a)^2/a*x^3*c^4*d+3/8*b/(b*x^2+a)^2/a^2*x^3*c^5+15/8/b^5/(b*x^2+a)^2*x*a^4*d^5-55/8/b^4/
(b*x^2+a)^2*x*a^3*c*d^4+35/4/b^3/(b*x^2+a)^2*x*a^2*c^2*d^3-15/4/b^2/(b*x^2+a)^2*x*a*c^3*d^2-5/8/b/(b*x^2+a)^2*
x*c^4*d+5/8/(b*x^2+a)^2*x/a*c^5-63/8/b^5*a^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d^5+175/8/b^4*a^2/(a*b)^(1/2)
*arctan(b*x/(a*b)^(1/2))*c*d^4-75/4/b^3*a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^2*d^3+15/4/b^2/(a*b)^(1/2)*arc
tan(b*x/(a*b)^(1/2))*c^3*d^2+5/8/b/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^4*d+3/8/a^2/(a*b)^(1/2)*arctan(b*x/
(a*b)^(1/2))*c^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^5/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.60851, size = 2195, normalized size = 11.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^5/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/240*(48*a^3*b^5*d^5*x^9 + 16*(25*a^3*b^5*c*d^4 - 9*a^4*b^4*d^5)*x^7 + 16*(150*a^3*b^5*c^2*d^3 - 175*a^4*b^4
*c*d^4 + 63*a^5*b^3*d^5)*x^5 + 10*(9*a*b^7*c^5 + 15*a^2*b^6*c^4*d - 150*a^3*b^5*c^3*d^2 + 750*a^4*b^4*c^2*d^3
- 875*a^5*b^3*c*d^4 + 315*a^6*b^2*d^5)*x^3 + 15*(3*a^2*b^5*c^5 + 5*a^3*b^4*c^4*d + 30*a^4*b^3*c^3*d^2 - 150*a^
5*b^2*c^2*d^3 + 175*a^6*b*c*d^4 - 63*a^7*d^5 + (3*b^7*c^5 + 5*a*b^6*c^4*d + 30*a^2*b^5*c^3*d^2 - 150*a^3*b^4*c
^2*d^3 + 175*a^4*b^3*c*d^4 - 63*a^5*b^2*d^5)*x^4 + 2*(3*a*b^6*c^5 + 5*a^2*b^5*c^4*d + 30*a^3*b^4*c^3*d^2 - 150
*a^4*b^3*c^2*d^3 + 175*a^5*b^2*c*d^4 - 63*a^6*b*d^5)*x^2)*sqrt(-a*b)*log((b*x^2 + 2*sqrt(-a*b)*x - a)/(b*x^2 +
a)) + 30*(5*a^2*b^6*c^5 - 5*a^3*b^5*c^4*d - 30*a^4*b^4*c^3*d^2 + 150*a^5*b^3*c^2*d^3 - 175*a^6*b^2*c*d^4 + 63
*a^7*b*d^5)*x)/(a^3*b^8*x^4 + 2*a^4*b^7*x^2 + a^5*b^6), 1/120*(24*a^3*b^5*d^5*x^9 + 8*(25*a^3*b^5*c*d^4 - 9*a^
4*b^4*d^5)*x^7 + 8*(150*a^3*b^5*c^2*d^3 - 175*a^4*b^4*c*d^4 + 63*a^5*b^3*d^5)*x^5 + 5*(9*a*b^7*c^5 + 15*a^2*b^
6*c^4*d - 150*a^3*b^5*c^3*d^2 + 750*a^4*b^4*c^2*d^3 - 875*a^5*b^3*c*d^4 + 315*a^6*b^2*d^5)*x^3 + 15*(3*a^2*b^5
*c^5 + 5*a^3*b^4*c^4*d + 30*a^4*b^3*c^3*d^2 - 150*a^5*b^2*c^2*d^3 + 175*a^6*b*c*d^4 - 63*a^7*d^5 + (3*b^7*c^5
+ 5*a*b^6*c^4*d + 30*a^2*b^5*c^3*d^2 - 150*a^3*b^4*c^2*d^3 + 175*a^4*b^3*c*d^4 - 63*a^5*b^2*d^5)*x^4 + 2*(3*a*
b^6*c^5 + 5*a^2*b^5*c^4*d + 30*a^3*b^4*c^3*d^2 - 150*a^4*b^3*c^2*d^3 + 175*a^5*b^2*c*d^4 - 63*a^6*b*d^5)*x^2)*
sqrt(a*b)*arctan(sqrt(a*b)*x/a) + 15*(5*a^2*b^6*c^5 - 5*a^3*b^5*c^4*d - 30*a^4*b^4*c^3*d^2 + 150*a^5*b^3*c^2*d
^3 - 175*a^6*b^2*c*d^4 + 63*a^7*b*d^5)*x)/(a^3*b^8*x^4 + 2*a^4*b^7*x^2 + a^5*b^6)]

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Sympy [B]  time = 6.5243, size = 614, normalized size = 3.13 \begin{align*} \frac{\sqrt{- \frac{1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right ) \log{\left (- \frac{a^{3} b^{5} \sqrt{- \frac{1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right )}{63 a^{5} d^{5} - 175 a^{4} b c d^{4} + 150 a^{3} b^{2} c^{2} d^{3} - 30 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d - 3 b^{5} c^{5}} + x \right )}}{16} - \frac{\sqrt{- \frac{1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right ) \log{\left (\frac{a^{3} b^{5} \sqrt{- \frac{1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right )}{63 a^{5} d^{5} - 175 a^{4} b c d^{4} + 150 a^{3} b^{2} c^{2} d^{3} - 30 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d - 3 b^{5} c^{5}} + x \right )}}{16} + \frac{x^{3} \left (17 a^{5} b d^{5} - 65 a^{4} b^{2} c d^{4} + 90 a^{3} b^{3} c^{2} d^{3} - 50 a^{2} b^{4} c^{3} d^{2} + 5 a b^{5} c^{4} d + 3 b^{6} c^{5}\right ) + x \left (15 a^{6} d^{5} - 55 a^{5} b c d^{4} + 70 a^{4} b^{2} c^{2} d^{3} - 30 a^{3} b^{3} c^{3} d^{2} - 5 a^{2} b^{4} c^{4} d + 5 a b^{5} c^{5}\right )}{8 a^{4} b^{5} + 16 a^{3} b^{6} x^{2} + 8 a^{2} b^{7} x^{4}} + \frac{d^{5} x^{5}}{5 b^{3}} - \frac{x^{3} \left (3 a d^{5} - 5 b c d^{4}\right )}{3 b^{4}} + \frac{x \left (6 a^{2} d^{5} - 15 a b c d^{4} + 10 b^{2} c^{2} d^{3}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**5/(b*x**2+a)**3,x)

[Out]

sqrt(-1/(a**5*b**11))*(a*d - b*c)**3*(63*a**2*d**2 + 14*a*b*c*d + 3*b**2*c**2)*log(-a**3*b**5*sqrt(-1/(a**5*b*
*11))*(a*d - b*c)**3*(63*a**2*d**2 + 14*a*b*c*d + 3*b**2*c**2)/(63*a**5*d**5 - 175*a**4*b*c*d**4 + 150*a**3*b*
*2*c**2*d**3 - 30*a**2*b**3*c**3*d**2 - 5*a*b**4*c**4*d - 3*b**5*c**5) + x)/16 - sqrt(-1/(a**5*b**11))*(a*d -
b*c)**3*(63*a**2*d**2 + 14*a*b*c*d + 3*b**2*c**2)*log(a**3*b**5*sqrt(-1/(a**5*b**11))*(a*d - b*c)**3*(63*a**2*
d**2 + 14*a*b*c*d + 3*b**2*c**2)/(63*a**5*d**5 - 175*a**4*b*c*d**4 + 150*a**3*b**2*c**2*d**3 - 30*a**2*b**3*c*
*3*d**2 - 5*a*b**4*c**4*d - 3*b**5*c**5) + x)/16 + (x**3*(17*a**5*b*d**5 - 65*a**4*b**2*c*d**4 + 90*a**3*b**3*
c**2*d**3 - 50*a**2*b**4*c**3*d**2 + 5*a*b**5*c**4*d + 3*b**6*c**5) + x*(15*a**6*d**5 - 55*a**5*b*c*d**4 + 70*
a**4*b**2*c**2*d**3 - 30*a**3*b**3*c**3*d**2 - 5*a**2*b**4*c**4*d + 5*a*b**5*c**5))/(8*a**4*b**5 + 16*a**3*b**
6*x**2 + 8*a**2*b**7*x**4) + d**5*x**5/(5*b**3) - x**3*(3*a*d**5 - 5*b*c*d**4)/(3*b**4) + x*(6*a**2*d**5 - 15*
a*b*c*d**4 + 10*b**2*c**2*d**3)/b**5

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Giac [A]  time = 1.13319, size = 459, normalized size = 2.34 \begin{align*} \frac{{\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b^{5}} + \frac{3 \, b^{6} c^{5} x^{3} + 5 \, a b^{5} c^{4} d x^{3} - 50 \, a^{2} b^{4} c^{3} d^{2} x^{3} + 90 \, a^{3} b^{3} c^{2} d^{3} x^{3} - 65 \, a^{4} b^{2} c d^{4} x^{3} + 17 \, a^{5} b d^{5} x^{3} + 5 \, a b^{5} c^{5} x - 5 \, a^{2} b^{4} c^{4} d x - 30 \, a^{3} b^{3} c^{3} d^{2} x + 70 \, a^{4} b^{2} c^{2} d^{3} x - 55 \, a^{5} b c d^{4} x + 15 \, a^{6} d^{5} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{2} b^{5}} + \frac{3 \, b^{12} d^{5} x^{5} + 25 \, b^{12} c d^{4} x^{3} - 15 \, a b^{11} d^{5} x^{3} + 150 \, b^{12} c^{2} d^{3} x - 225 \, a b^{11} c d^{4} x + 90 \, a^{2} b^{10} d^{5} x}{15 \, b^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^5/(b*x^2+a)^3,x, algorithm="giac")

[Out]

1/8*(3*b^5*c^5 + 5*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 150*a^3*b^2*c^2*d^3 + 175*a^4*b*c*d^4 - 63*a^5*d^5)*arct
an(b*x/sqrt(a*b))/(sqrt(a*b)*a^2*b^5) + 1/8*(3*b^6*c^5*x^3 + 5*a*b^5*c^4*d*x^3 - 50*a^2*b^4*c^3*d^2*x^3 + 90*a
^3*b^3*c^2*d^3*x^3 - 65*a^4*b^2*c*d^4*x^3 + 17*a^5*b*d^5*x^3 + 5*a*b^5*c^5*x - 5*a^2*b^4*c^4*d*x - 30*a^3*b^3*
c^3*d^2*x + 70*a^4*b^2*c^2*d^3*x - 55*a^5*b*c*d^4*x + 15*a^6*d^5*x)/((b*x^2 + a)^2*a^2*b^5) + 1/15*(3*b^12*d^5
*x^5 + 25*b^12*c*d^4*x^3 - 15*a*b^11*d^5*x^3 + 150*b^12*c^2*d^3*x - 225*a*b^11*c*d^4*x + 90*a^2*b^10*d^5*x)/b^
15